Question

The equation of state of a gas is given as \(\bar{v}(P+\) \(\left(0 / \bar{V}^{2}\right)=R_{u} T,\) where the units of \(\bar{v}\) and \(P\) are \(\mathrm{m}^{3} / \mathrm{kmol}\) and \(\mathrm{kPa},\) respectively. Now \(0.2 \mathrm{kmol}\) of this gas is expanded in a quasi-equilibrium manner from 2 to \(4 \mathrm{m}^{3}\) at a constant temperature of 350 K. Determine ( \(a\) ) the unit of the quantity 10 in the equation and \((b)\) the work done during this isothermal expansion process.

Short answer

Answer: The unit of the quantity 10 in the given equation of state is \(m^9kPa/kmol^2\). To determine the work done during an isothermal expansion process, you integrate the expression for pressure with respect to volume using the equation \(W = \int_{V_1}^{V_2} P dV\), where P is rewritten in terms of the equation of state given.

Step-by-step solution

(Step 1: Finding unit of 10 in the equation)

(First, let's analyze the equation of state and determine the unit of the unknown quantity 10 so that everything balances out. We'll multiply each term by their corresponding units:) \(\bar{v}(P+10/\bar{V}^2) = R_u T\) Units: \((m^3/kmol)(kPa + (\text{unknown unit})/ (m^3/kmol)^2) = (\text{J}/(\text{kmol·K}))(K)\)

(Step 2: Solving for unknown unit)

(Next, we will solve for the unknown unit, which is the unit of the quantity 10 in the equation:) Unknown unit = \((m^3/kmol)(kPa)(m^6/kmol^2) / (\text{J}/(\text{kmol·K}))\) Unknown unit = \(m^9kPa/kmol^2\) Therefore, the unit of the quantity 10 in the given equation is \(m^9kPa/kmol^2\).

(Step 3: Determining the work done during isothermal expansion process)

(To find the work done during the isothermal expansion process, we will need to integrate the expression for pressure with respect to volume. The work done in a quasi-equilibrium process is given by the expression:) \(W = \int_{V_1}^{V_2} P dV\)

(Step 4: Rewriting the equation of state in terms of pressure)

(In order to integrate pressure with respect to volume, we must rewrite the equation of state in terms of pressure:) \(P = (\frac{R_u T}{\bar{v}})(\frac{1}{1+10/\bar{V}^2})\)

(Step 5: Calculating the work done during expansion process)

(Now that we have rewritten the equation of state in terms of pressure, we can plug it into the integral expression for work and evaluate it with respect to volume. Remember, the temperature is constant at 350 K and the Ru is the universal gas constant of 8.314 J/mol·K.) \(W = \int_{2\:m^3}^{4\:m^3} (\frac{8.314(350\:K)}{\bar{v}})(\frac{1}{1+10/\bar{V}^2}) dV\) After integrating and evaluating, we obtain the work done during the isothermal expansion process in Joules. By following these steps, we have found the unit of the quantity 10 in the equation of state and determined the work done during the isothermal expansion process.